The meta properties of finite ring homomorphisms.

Main results

Let R be a commutative ring, S is an R-algebra, M be a submonoid of R.

• finite_localizationPreserves : If S is a finite R-algebra, then S' = M⁻¹S is a finite R' = M⁻¹R-algebra.
• finite_ofLocalizationSpan : S is a finite R-algebra if there exists a set { r } that spans R such that Sᵣ is a finite Rᵣ-algebra.

theorem RingHom.finite_stableUnderComposition : StableUnderComposition @Finite

theorem RingHom.finite_respectsIso : RespectsIso @Finite

theorem RingHom.finite_containsIdentities : ContainsIdentities @Finite

theorem RingHom.finite_isStableUnderBaseChange : IsStableUnderBaseChange @Finite

theorem RingHom.finite_localizationPreserves : LocalizationPreserves @Finite

If S is a finite R-algebra, then S' = M⁻¹S is a finite R' = M⁻¹R-algebra.

theorem RingHom.localization_away_map_finite (R S R' S' : Type u) [CommRing R] [CommRing S] [CommRing R'] [CommRing S'] [Algebra R R'] (f : R →+* S) [Algebra S S'] (r : R) [IsLocalization.Away r R'] [IsLocalization.Away (f r) S'] (hf : f.Finite) : (IsLocalization.Away.map R' S' f r).Finite

theorem RingHom.finite_ofLocalizationSpan : OfLocalizationSpan @Finite

S is a finite R-algebra if there exists a set { r } that spans R such that Sᵣ is a finite Rᵣ-algebra.